Intercept, of regression line

Table 2. Slopes and intercepts of regression lines for several infant cereals and one breakfast cereal...

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

Legend No number of measurement. Cone concentration in fig, CN"/100 ml Absorb absorbance [AU] slope slope of regression line t CV intercept see slope res. std. dev. residual standard deviation Srts -n number of points in regression LOD limit of detection LOQ limit of quantitation measurements using a 2-fold higher sample amount and 5-cm cuvettes—i.e., measured absorption 0. .. 0.501 was divided by 10. 

According to the ideal gas law, the pressure of a gas is directly proportional to the absolute temperature, that is, P = (nRJV)T. This linear relationship can be represented by P = mT + b, where m is the slope, equal to (nR/V), and b is the intercept of the line, which is zero in the absence of any imprecision in the data. The slope and intercept may thus be determined by linear regression of the four datapoints versus the model equation P = mT + b. 

For each of the solutions used in the calibration, represent the sensor s response (in mV) versus the log Ok+ (or log aNH4+). Determine visually the lineal range for the sensors response and use the linear regression in order to obtain values for (a) correlation coefficient, (b) slope and (c) intercept of the line. 

Figure 8. Plot of CaO against Na20 for the chondrule mesostases in Semarkona and Krymka. The diagonal line is a regression line through the data for mesostases which luminesce and the broken lines refer to 2 a uncertainties on the intercepts of this line. The dotted line refers to the theoretical line for plagioclase. (Reprinted by permission from Ref. 11. Copyright 1987 Lunar and Planetary Institute.)...

Through linear regression analysis of the data the slope and intercept of the line at 1 = 3.0 were — 23.024 and 54.142, respectively. The correlation coefficient was 0.9949 and the mean deviation about the regression line was 0.6818. Similarly for the line at 1 = 1.5 the slope and the intercept were —17.27 and 30.145, respectively, while the correlation coefficient and the mean deviation about the regression line were 0.9968 and 0.3120, respectively. At I = 0.6 the slope was — 4.775 while the intercept was 10.678. The correlation coefficient was 0.9664 and the mean deviation... 

Figure 2 shows the good correlation between the obtained results. The uncertainties intercept the regression line and the linear coefficient of regression was acceptable. It confirms linearity. 

Figure 2.2 Plot of regression line for a single categorical covariate (sex) with two levels (males and females). The effect of the categorical variable is to shift the model intercept.

Before the equation of the line calculated using the zero intercept model is employed to evaluate unknowns, it must be tested to determine if the model is adequate to describe the experimental data. Regression analysis tables are constructed prior to testing the statistical validity of the assumption that the intercept of the line is zero. The format for calculation of the regression analysis tables is shown in Table Ill-a and the analyses of the Table I data are shown in Table Ill-b. 

A second objection to using the line of regression of y on x, as calculated in Sections 5.4 and 5.5, in the comparison of two analytical methods is that it also assumes that the error in the y-values is constant. Such data are said to be homoscedastic. As previously noted, this means that all the points have equal weight when the slope and intercept of the line are calculated. This assumption is obviously likely to be invalid in practice. In many analyses, the data are heteroscedastic, i.e. the standard deviation of the y-values increases with the... 

Theil s method determines the slope of a regression line as the median of the slopes calculated from selected pairs of points the intercept of the line is the median of the intercept values calculated from the slope and the coordinates of the individual points. 

Chapter 6. For the purposes of demonstrating this relationship it is sufficient to say that the values of the logarithm of a reciprocal concentration (log l/Q in eqn (1.2) are obtained by multiplication of the values by a coefficient (1.039) and the addition of a constant term (—0.442). The equation is shown in graphical form (Fig. 1.2) the slope of the fitted line is equal to the regression coefficient (1.039) and the intercept of the line with the zero point of the x-axis is equal to the constant (—0.442). 

There are some measures of special importance. The slope of the regression line, the intercept of the regression line (intersection of regression line with the y-axis), the end of serial processing ES (defined by the first task > v55 with falling median reaction time in the next task). 

Figure B2.4.2. Eyring plot of log(rate/7) versus (1/7), where Jis absolute temperature, for the cis-trans isomerism of the aldehyde group in fiirfiiral. Rates were obtained from tln-ee different experiments measurements (squares), bandshapes (triangles) and selective inversions (circles). The line is a linear regression to the data. The slope of the line is A H IR, and the intercept at 1/J = 0 is A S IR, where R is the gas constant. A and A are the enthalpy and entropy of activation, according to equation (B2.4.1)...

The most commonly used form of linear regression is based on three assumptions (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x. Because we assume that indeterminate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

The determination of errors in the slope a and the intercept b of the regression line together with multiple and curvilinear regression is beyond the scope of this book but references may be found in the Bibliography, page 156. 

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